In this work, we study methodical decomposition of an undirected, unweighted complete graph ($K_n$ of order $n$, size $m$) into minimum number of edge-disjoint trees. We find that $x$, a positive integer, is minimum and $x=\lceil\frac{ n}{ 2}\rceil$ as the edge set of $K_n$ is decomposed into edge-disjoint trees of size sequence $M = \{ m_1,m_2,...,m_x\}$ where $m_i\le(n-1)$ and $\Sigma_{ i=1}^{ x} m_i$ = $\frac{ n(n-1)}{ 2}$. For decomposing the edge set of $K_n$ into minimum number of edge-disjoint trees, our proposed algorithm takes total $O(m)$ time.
Categories: cs.DM